Class 6a: Time Series

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.title[
# Class 6a: Time Series
]
.author[
### Business Forecasting
]

---

</style>

---

###Roadmap:

1. Components of time series
2. Patterns of correlation in time series
3. Simple forecasting methods
4. Evaluating forecasts
5. Time series decomposition
6. Forecasting with time series decomposition

---
### Example 
- Suppose you wonder if you should go into the wedding business.
- You need to predict whether there is potential for work
- So you look at evolution in the number of weddings across years

#### Heterosexual Marriages in Mexico
<img src="data:image/png;base64,#C_6_slides_a_files/figure-html/unnamed-chunk-1-1.png" width="100%" />

What patterns can you identify in that time series?
---
### Same Sex Marriages in Mexico

--
Going into gay marriage business is probably a better idea!

---
### Components

1. **Trend** - long term change in the level of data, positive or negative.
  - If flat, we call the data stationary
  - Formally, the mean, variance, and autocorrelation does not depend on time

---

### Heterosexual Marriages in Mexico

---
### Components

1. **Trend** - long term change in the level of data, positive or negative.
  - If flat, we call the data stationary
  - Formally, the mean, variance, and autocorrelation does not depend on time

2. **Seasonal pattern**: Variation in level that repeats at the same time each period
  - If there is seasonality, data is not stationary

---

### Heterosexual Marriages in Mexico

---
### Components

1. **Trend**: Long term change in the level of data, positive or negative.
  - If flat, we call the data stationary
  - Formally, the mean, variance, and autocorrelation does not depend on time

2. **Seasonal pattern**: Variation in level that repeats at the same time each period
  - If there is seasonality, data is not stationary

3. **Cyclical pattern**: Wavelike upward and downward movements along the trend. Not always the same length, not always the same time of year
  - Different from seasonality which always happens at the same time and has same length
  - Often related to business cycles

---

### Heterosexual Marriages in Mexico

---
### Components

1. **Trend**: Long term change in the level of data, positive or negative.
  - If flat, we call the data stationary
  - Formally, the mean, variance, and autocorrelation does not depend on time

2. **Seasonal pattern**: Variation in level that repeats at the same time each period
  - If there is seasonality, data is not stationary

4. **Random components**:  Can't be attributed to other parts of the model. The most difficult to predict

---

### Heterosexual Marriages in Mexico

---
### Some other examples

---

---

---

---
### Autocorrelation

- Can past values predict future values?
- How strong is "memory" in the data?

--
- Yes, if they are correlated

--
- We will measure **Autocorrelation**: 
    - Are values in previous period correlated with values in the next period?
    - So between `$y_t$` and `$y_{t-1}$`, or `$y_t$` and `$y_{t-2}$` etc
    
`$$\hat{\rho}_k=\frac{\sum^{n}_{t=k+1}(y_{t}-\bar{y})(y_{t-k}-\bar{y})}{\sum^{n}_{t=1}(y_t-\bar{y})^2}$$`

```
## # A tsibble: 384 x 5 [1M]
## Month M Lag1_M Lag2_M Lag3_M
## <mth> <dbl> <dbl> <dbl> <dbl>
## 1 1993 Jan 54850 NA NA NA
## 2 1993 Feb 54271 54850 NA NA
## 3 1993 Mar 55350 54271 54850 NA
## 4 1993 Apr 52268 55350 54271 54850
## 5 1993 May 59671 52268 55350 54271
## 6 1993 Jun 47557 59671 52268 55350
## 7 1993 Jul 54503 47557 59671 52268
## 8 1993 Aug 51534 54503 47557 59671
## 9 1993 Sep 46000 51534 54503 47557
## 10 1993 Oct 51590 46000 51534 54503
## # ℹ 374 more rows
```

---
We can calculate the values for marriage data:

|    |          |          |          |          |          |          |
|:---|---------:|---------:|---------:|---------:|---------:|---------:|
|lag | 1.0000000| 2.0000000| 3.0000000| 4.0000000| 5.0000000| 6.0000000|
|acf | 0.3070272| 0.4924005| 0.3022587| 0.2484164| 0.2805866| 0.1646544|

And plot the Autocorrelation Function (ACF) on a correlogram:

- Why high values at 12 and 24 lag?

---
### Some other examples:

- If there is a trend there will be positive autocorrelation

---

- Shock persists for a long time
- If stationary, shocks should not persist, autocorrelation should decay quickly to 0

---

![](data:image/png;base64,#C_6_slides_a_files/figure-html/unnamed-chunk-16-1.png)

---
### Autocorrelation

- How do we know that the correlation is significant and not just sampling randomness?
- Test:

- `$H_0: \rho_k=0$` or data is white noise
  - `$H_A: \rho_k\neq 0$`

--
- What is **White Noise**?

<img src="data:image/png;base64,#C_6_slides_a_files/figure-html/unnamed-chunk-17-1.png" width="100%" />
**White noise** data is uncorrelated across time, with zero mean and constant variance
 - It is stationary!

---
### White Noise

Autocorrelation of white noise
<img src="data:image/png;base64,#C_6_slides_a_files/figure-html/unnamed-chunk-18-1.png" width="100%" />

---
### Test

- Intuitively:
  1. We will calculate test statistic 
  2. Figure out how likely to obtain such value if data was White Noise
  
--
- If test statistic is big, it's unlikely to come from White Noise, so we reject null

`$$t_{test}=\frac{\hat{\rho}_k-0}{1/\sqrt{n-k}}$$`

- Compare it to t distribution with `$t_{n-k}$` degrees of freedom

--
- Rule of thumb for larger datasets: reject at 95% if:

`$$|\hat{\rho}_k|>\frac{2}{\sqrt{n}}$$`

---
### Confidence bands

- We can compute confidence bands such that if `$\hat{\rho}_k$` is within these bands, it's not significant.

- In our data on straight marriage, n=360

- If data is white noise, autocorrelations should not cross 0.1054

- The more observation you have, the better you are at detecting autocorrelation
---
### Gay marriages

--
- Is there a way to transform the data to remove the trend?

---
### First differencing

- Take the first differences

`$$\Delta y_t=y_t-y_{t-1}$$`
- First differences approximate how much data grows in each period
- If trend is linear, this variable should have more or less constant mean

---
### First differencing

```
## # A tsibble: 178 x 3 [1M]
## Month M Diff_M
## <mth> <int> <int>
## 1 2010 Mar 94 NA
## 2 2010 Apr 72 -22
## 3 2010 May 60 -12
## 4 2010 Jun 68 8
## 5 2010 Jul 61 -7
## 6 2010 Aug 69 8
## 7 2010 Sep 52 -17
## 8 2010 Oct 69 17
## 9 2010 Nov 47 -22
## 10 2010 Dec 97 50
## # ℹ 168 more rows
```

---

Is transform data stationary?
<img src="data:image/png;base64,#C_6_slides_a_files/figure-html/unnamed-chunk-22-1.png" width="100%" />

--
- Does it have constant mean?

--
- What about constant variance?

--
- What about autocorrelation?

---
<img src="data:image/png;base64,#C_6_slides_a_files/figure-html/unnamed-chunk-23-1.png" width="100%" />

---

### Simple forecasting methods

**Naive Model**

The simplest way to forecast is to assume that it will be the same as previous period

- One step forecast: `$\hat{y}_{T+1|T}=y_T$`

--
- h-step forecast: `$\hat{y}_{T+h|T}=y_T$`

---
### Simple forecasting methods

What is the confidence interval for such prediction?

- We need to know the variance of the forecast error

--
- What is **Forecast Error**?

`$$e_t=y_{T+h}-\hat{y}_{T+h|T}$$`
--

- It's the difference between what we forecasted based on our model and what actually happened once we observe this data point
- Also known as out-of-sample error, because the forecasted point was not in the estimation sample
- We only used observations up to point T when estimating this model!

--
- Different from **Fitted Residuals**!

`$$u_t=y_{t}-\hat{y}_{t}$$`
These are fitted residuals for observations that we used in estimation.

---

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---

### Simple forecasting methods

- In the simplest naive model, the one step ahead residuals and forecast errors are similar.

- So we can approximate the standard deviation of `$e_t$` with standard deviation of `$u_t$` in this naive model.

- Let `$\sigma_h$` be the h-step forecast error.

- We will assume: `$$\sigma_1=\sigma_u$$` so the standard deviation of the one step ahead forecast is the same as the standard deviation of the residuals

- This gives us the following confidence interval for one step ahead error:

`$$CI_{95}=\hat{y}_{T+1|T} \pm 1.96\hat{\sigma}_u$$`
---
### Simple forecasting methods

For longer horizon, forecast error in naive forecast is:

- Let `$\sigma_h=\sigma_u\sqrt{h}$` be the sd of h-step forecast error, and `$$CI_{95}=\hat{y}_{T+h|T} \pm 1.96\hat{\sigma}_u\sqrt{h}$$`

---
### Simple forecasting methods

```
## [1] "error standard deviation: 13903.9441194794"
```

```
##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 385          50519 32723.65 68314.35 23303.36 77734.64
```

---
### Simple forecasting methods

**Seasonal naive**

We can make it slightly more elaborate by assuming it's the same value as in the last same season:

`$\hat{y}_{T+1|T}=y_{m(T+1)}$`

- m(T+1) is the last time period with the same season as T+1

```
## [1] "error standard deviation: 6512.04123595939"
```

---

### Simple forecasting methods

- At one step ahead, the confidence interval is the same:

`$$CI_{95}=\hat{y}_{T+1|T} \pm 1.96\hat{\sigma}_u$$`

- For longer horizon, forecast error is slightly different:

- Let `$\sigma_h$` be the h-step forecast error sd
- Let `$k$` be the number of seasonal cycles in the forecast prior to forecast time
  - If it's the first January since time T, no full cycle has passed k=0, k+1=1
  - If it's the second January since time T, 1 full cycle has passed k=1, k+1=2

`$$CI_{95}=\hat{y}_{T+h|T} \pm 1.96\hat{\sigma}_u\sqrt{k+1}$$`
---

### Simple forecasting methods

```
## [1] "error standard deviation: 6512.04123595939"
```

--
- Why the interval is smaller than in the previous case?

--
- The forecast errors are smaller
- So the standard deviation of errors is smaller!

---

### Simple forecasting methods

**Simple Average**

We can also just take an average of the time series and make it our prediction:

`$\hat{y}_{T+1|T}=\bar{y}_T=\frac{\sum_{t\leq T} y_t}{T}$`

---

### Simple forecasting methods

- At one step ahead, the confidence interval is the same:

`$$CI_{95}=\hat{y}_{T+1|T} \pm 1.96\hat{\sigma}_u$$`

- For longer horizon, forecast error is slightly different:

- Let `$\sigma_h=\sigma_u\sqrt{h+\frac{1}{T}}$` be the h-step forecast error sd

`$$CI_{95}=\hat{y}_{T+h|T} \pm 1.96\hat{\sigma}_u\sqrt{h+\frac{1}{T}}$$`

- Generally, average value across 20 years is not a good prediction

---

### Moving average

Consider an average of the last `$k$` observations:

`$$MA(k)_t: \frac{\sum^k_{j=1} y_{t+1-j} }{k}=\frac{y_{t}+y_{t-1}..+y_{t+1-k}}{k}$$`

- How many? Usually equal to number of seasons, so the seasonal variation smoothed out

--
- As we will see later, this is more useful in identifying trend and cycle components

**12 months**

---

**4 Months**

---

**3 years**

---

### Evaluating forecasts

Which of the forecasts was the best?

--
- There is couple of ways to evaluate the forecast accuracy
- They all have advantages and disadventages
- General idea: how close the forecast was to the observed value
- You always use OUT-OF-SAMPLE errors, not fitted residuals
---

### Mean Error

`$$ME=\frac{\sum^{T-h}_{t=1}(y_{t+h}-\hat{y}_{t+h|t})}{T-h}$$`
- This is the average of forecast error

--
- Can tell us which direction is the bias

- You can test for the existence of bias with a usual t test:

- `$\small H_0:E(e_t)=0$`
  - `$\small H_A:E(e_t) \neq 0$` (or inequality)

- Test statistic and the null distribution:
  `$$\small T_{test}=\frac{\bar e-0}{\frac{\hat{\sigma}_e}{\sqrt n}} \sim t_{n-1}$$`

--
- Positive and negative values can add up to 0
- So even if errors are large, but symmetric, this measure will be close to 0

---
### Mean Error

My estimation sample is up to December 2009. I am trying to predict values in 2010

---
### Mean Error:

To calculate the mean error at h=1, I would repeat this repetitively increasing my estimation sample.

- Ex: Estimate based on data up to January 2010, and predict February 2010, compute error
- Next: Estimate based on data up to February 2010, and predict March 2010, compute error
- And so on... until T-1. 
- Then compute average of these errors

```
## # A tibble: 3 × 11
## .model SS .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
## <chr> <lgl> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 Mean FALSE Test -4822. 12980. 12050. -17.1 29.0 4.19 3.51 -0.5
## 2 Naïve FALSE Test -13170. 17851. 13170. -35.3 35.3 4.58 4.83 -0.5
## 3 Seasonal naïve FALSE Test -5940. 5951. 5940. -12.8 12.8 2.06 1.61 -0.5
```
- If the error is negative, we overestimate!

---
### Mean Absolute Error

`$$MAE=\frac{\sum^{T-h}_{t=1}|y_{t+h}-\hat{y}_{t+h|t}|}{T-h}$$`

- Similar, but we take absolute value of errors. So they don't cancel out!
- This measure is **always** positive 
- But we can't say whether we underpredict or overpredict

- Now clearly seasonal is the best

---
### Mean Percentage Error

`$$MPE=\frac{\sum^{T-h}_{t=1}(y_{t+h}-\hat{y}_{t+h|t})/|y_{t+h}|}{T-h}$$`
- Answers the question:
  - on average, my forecast is x% wrong (deviates x% from true value)
  
--
- It's unitless, so I can compare forecasts of different measures
  - EG: comparing forecast of inflation vs exports

--
- But again, negative and positive can cancel out...

- So average forecast is again performing well!

---
### Mean Absolute Percentage Error

`$$MAPE=\frac{\sum^{T-h}_{t=1}|y_{t+h}-\hat{y}_{t+h|t}|/|y_{t+h}|}{T-h}$$`
- Similar as before, but we take the absolute value

---
### Squared Errors

- Mean Squared Errors
`$$MSE=\frac{\sum^{T-h}_{t=1}(y_{t+h}-\hat{y}_{t+h|t})^2}{T-h}$$`

- Root Mean Squared Errors
`$$RMSE=\sqrt{\frac{\sum^{T-h}_{t=1}(y_{t+h}-\hat{y}_{t+h|t})^2}{T-h}}$$`
--

- If we take a sqaure instead of absolute value, we penalize more big deviations
- Then we need to take square root to get the right units back

---

### Time series decomposition

- Helps in analyzing the patterns in the time series data
- Sometimes used for forecasting

**Multiplicative Decomposition**: Assume time series is a product of 4 elements:

`$$y_t=S_tT_tC_tR_t$$`

--
**Goal**: Identify the elements of the time series:

1. Seasonality `$S_t$`
2. Trend `$T_t$`
3. Cycles `$C_t$`
4. Irregular/Reminder `$R_t$`

--
- Two notes:
  - We will often ignore irregular components
  - Some methods don't distinguish between Trend and Cycles

---
### Seasonality

- How would you identify which variations in time series are due to seasonality?

--
- **Idea**: 
  1. Eliminate seasonal variation - deseason
  2. Compare the actual series to the one without seasonal variation
  3. The difference is due to seasonality!

--
- How to eliminate seasonal variation?

--
- We will use (Centred) Moving Averages for smoothing.

---
### Seasonality

#### Moving averages for smoothing

- Why moving average smoothes out seasonal variation?

--
- It averages out variation over some period of time

--
- In some seasons we have more weddings, in some seasons we have less wedding. On average these positive and negative seasonalities will average out.

--
- Over which period should we take average?

---
### Seasonality

- Suppose I take average over 5 months. 
- Note that this time the period in focus is at the center
- I look at `$y_t$`, two observations before it and two observations after it! 
- So the closest observations to `$y_t$`

`$$\small MA(5)_t: \frac{\sum^{2}_{j=-2} y_{t+j} }{5}=\frac{y_{t+2}+y_{t+1}+y_t+y_{t-1}+y_{t-2}}{5}$$`

--
- If I take only 5, I don't average over all seasons!
- Sometimes I capture more high seasons but not low seasons, so seasonality persists

---
### Seasonality

- Suppose I take average over 12 seasons
- Now each average is over all months (seasons)
- We eliminated seasonal variation

`$$\small MA(12)_t: \frac{\sum^{5}_{j=-6} y_{t+j} }{12}=\frac{y_{t+5}+y_{t+4}..+y_t+..+y_{t-5}+y_{t-6}}{12}$$`

---
### Seasonality

--
- Mathematical caveat
  - Since the number of periods is even (12), our main observations is not really at the center
  - We can have 5 obs before and 6 after
  - Or 6 obs before and 5 after

--
**Centered** Moving Average

- Or we can have both!

--
- Calculate moving average both ways and then take the average of the two.

`$$CMA(12)_t=(\frac{\sum^{5}_{j=-6} y_{t+j} }{12}+\frac{\sum^{6}_{j=-5} y_{t+j} }{12})/2$$`

--
- Note that we lose some data at the end and at the beginning.

---

- What was the "seasonality" in terms of Covid?

![My Plot](data:image/png;base64,#covid_average.jpg)

--
- Less testing on weekends

--
- Seasonality was by the day of the week

--
- So we take 7 days average to smooth it out

---
### Seasonality

- We achieved first step, we have a series without seasonal variation
- So how we identify which parts are due to seasonality?

--
**Seasonal indices**

- Compare actual data to data without seasons

--
- January 2010 seasonal factor would be
  `$$SF_{January, 2010}=Y_{January,2010}/CMA_{January,2010}$$`

---

<img src="data:image/png;base64,#C_6_slides_a_files/figure-html/unnamed-chunk-49-1.png" width="100%" />
 `$$SF_{t}=Y_{t}/CMA_{t}$$`

---
### Seasonality
**Seasonal indices**

- We assume seasonal indices are the same across the time, so we just take the average of all of them for each season:

`$$SF_{January}=\sum_{year}Y_{January,year}/CMA_{January,year}$$`

--
.pull-left[

```
##    Month Seasonal_index
## 1      1      0.8588455
## 2      2      1.3476408
## 3      3      1.0110684
## 4      4      0.9325319
## 5      5      1.0081259
## 6      6      0.8554597
## 7      7      0.9508683
## 8      8      0.9308511
## 9      9      0.8004292
## 10    10      0.9499117
## 11    11      0.9841737
## 12    12      1.3700936
```
]
.pull-right[

- In January, we have 13.5% less weddings than yearly average
- In December, we have on average 38% more weddings than yearly average
- in June, we have 14.3% less weddings than yearly average
]
---
### Seasonality

- They should average to 1
  - Because they represent how much they deviate from average in a given season
- (or in other words) They should add up to the number of seasons!
`$$\sum^S_{s=1}SF_s=S$$`
--
- If you don't know one index, you can identify it from the sum

---

### Trend

- We isolated seasonality
- Now that our time series is not contaminated by seasonal variation, we can identify the trend

**Assumption:** Trend is linear

- We are trying to find a line that best approximate the *deseasoned data*

--
- That's what a linear regression do!

--
- My outcome is the deseasoned time series values
- My predictor is time

--
`$$CMA_t=a+bt+e_t$$`

- We find `$a$` and `$b$` by OLS

--
- Our predicted trend at time t is:

`$$T_t=\hat{a}+\hat{b}t$$`

---
### Straight marriages

```
## 
## Call:
## lm(formula = trend ~ Time, data = a)
## 
## Coefficients:
## (Intercept)         Time  
##    59865.02       -57.96
```

---
### Same-sex marriages

```
## 
## Call:
## lm(formula = trend ~ Time, data = a)
## 
## Coefficients:
## (Intercept)         Time  
##     -20.416        2.989
```

---
### Cyclical element

- Cyclical element is the upward and downward movements around the trend in the deseasoned data

--
- Divide the centered moving average (deseasoned time series) by the trend value

`$$C_t=\frac{CMA_t}{T_t}$$`

---
### Multiplicative Decomposition

- What about the irregular component?
- We will assume it's one, unless someone tells us there will be some shock

- Once we identified all the elements, we can make predictions for the original variable using the model:

`$$y_t=S_tT_tC_tR_t$$`

---
### Prediction

What will be the marriage rate in January 2023 (t_{max}+1)?

- What is my `$S_{t_{max}+1}$`

--
  - `$S_{t_{max}+1}$` for January is:  0.865

--
- What is my `$T_{t_{max}+1}$`?

--
  - Formula: `$60304.43-61.46*361=38117.37$`
  - January 1993 is t=1, February 1993 is t=2 ... January 2023 is t=361

--
- What is my `$C_{t_{max}+1}$`?

--
  - Hardest to predict
  - Assume it's the same as last available one: `$C_{t_{max}-6}=1.0876$`

--
- What is my `$R_{t_{max}+1}$`?

--
  - We don't expect anything crazy to happen so `$R_{t_{max}+1}=1$`

--
  - Putting it all together:
  `$$\hat{y}_{t_{max}+1}=S_{t_{max}+1}T_{t_{max}+1}C_{t_{max}+1}R_{t_{max}+1}=35859.83$$`

---
### Prediction

**Confidence Interval**

**Step 1** Find interval bands from trend regression
  - Just use the standard formula
  
`$$\hat{T}_{t_{max}+1} \pm t_{\alpha/2,n-2}\sqrt{\hat{\sigma_T}^2(1+\frac{1}{n}+\frac{(t_{max}+1-\bar t)^2}{\sum^n_{t=1}(t-\bar t)^2})}$$`
  - t_{max} is the number of the last observation
  - `$T_{t_{max}+1}$` is the trend prediction 
  - `$\hat{\sigma}_T=\frac{SSE}{n-2}$` is the st.dev of residuals from the linear regression.

```
##       fit      lwr      upr
## 1 38941.4 32049.85 45832.94
```

**Step 2** Mulitply these bands by the seasonal and cyclical component

`$$CI_{95}=(31187*\underbrace{1.0876}_{C_{t_{max}+1}}*\underbrace{0.865}_{S_{t_{max}+1}}, 45050*\underbrace{1.0876}_{C_{t_{max}T+1}}*\underbrace{0.865}_{S_{t_{max}+1}})=(29339.92, 42381.87)$$`
- This is under the assumption there is little uncertainty about seasonal and cyclical component

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